JHEP06(2014)032 Published for SISSA by Springer Received: May 7, 2014 Accepted: May 14, 2014 Published: June 6, 2014 The two-loop master integrals for q ¯ q → VV Thomas Gehrmann, a Andreas von Manteuffel, b Lorenzo Tancredi a and Erich Weihs a a Physik-Institut, Universit¨ at Z¨ urich, Wintherturerstrasse 190, CH-8057 Z¨ urich, Switzerland b PRISMA Cluster of Excellence & Institute of Physics, Johannes Gutenberg University, 55099 Mainz, Germany E-mail: [email protected], [email protected], [email protected], [email protected]Abstract: We compute the full set of two-loop Feynman integrals appearing in massless two-loop four-point functions with two off-shell legs with the same invariant mass. These integrals allow to determine the two-loop corrections to the amplitudes for vector boson pair production at hadron colliders, q ¯ q → VV , and thus to compute this process to next-to- next-to-leading order accuracy in QCD. The master integrals are derived using the method of differential equations, employing a canonical basis for the integrals. We obtain analytical results for all integrals, expressed in terms of multiple polylogarithms. We optimize our results for numerical evaluation by employing functions which are real valued for physical scattering kinematics and allow for an immediate power series expansion. Keywords: QCD Phenomenology, Hadronic Colliders ArXiv ePrint: 1404.4853 Open Access,c The Authors. Article funded by SCOAP 3 . doi:10.1007/JHEP06(2014)032
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JHEP06(2014)032
Published for SISSA by Springer
Received: May 7, 2014
Accepted: May 14, 2014
Published: June 6, 2014
The two-loop master integrals for qq → V V
Thomas Gehrmann,a Andreas von Manteuffel,b Lorenzo Tancredia and Erich Weihsa
aPhysik-Institut, Universitat Zurich,
Wintherturerstrasse 190, CH-8057 Zurich, SwitzerlandbPRISMA Cluster of Excellence & Institute of Physics, Johannes Gutenberg University,
Expanding in ε, the canonical form ensures full decoupling of the differential equa-
tions (3.22) order by order in ε. The integration up to a boundary term becomes trivial
and can be carried out entirely algebraically.
4 Integration and boundary conditions
We consider the full system of differential equations for all 75 master integrals in a uniform
manner. Our normalization is such that the solutions for our master integrals have a Taylor
expansion,
~m(ε;x, z) =∞∑i=0
~m(i)(x, z)εi , (4.1)
where the weight 0 contributions start at ε0. We solve the full vector of coefficient functions
~m(i) order by order in ε up to and including weight 4.
For master integrals depending on z we choose to integrate the partial differential
equation in z for fixed x implied by (3.22). This gives us the solution up to a function of
x, which needs to be fixed by additional constraints discussed below. For master integrals
independent on z we integrate the partial differential equation in x, which determines the
solution up to a constant. It is obvious that this procedure naturally leads to iterated
integrals. The d ln form of the differential equations ensures that the iterated integrals can
be expressed in terms of Goncharov’s multiple polylogarithms
G(w1, w2, · · · , wn; z) ≡∫ z
0dt
1
t− w1G(w2, · · · , wn; t) , (4.2)
G(0, · · · , 0︸ ︷︷ ︸n
; z) ≡ 1
n!lnn z . (4.3)
– 14 –
JHEP06(2014)032
Here, the wi are complex rational functions of the indeterminants. To handle non-linear
letters we also employ generalized weights [59].
G([f(o)], w2, · · · , wn; z) =
∫ z
0dtf ′(t)
f(t)G(w2, · · · , wn; t) , (4.4)
where f(o) is an irreducible rational polynomial and o is a dummy variable.
In order to fix the boundary terms we use two ingredients. For some of the simplest
integrals, namely a small number of tadpole, bubble and triangle integrals, we use their
known analytic solutions from the literature [18, 60]. For all other integrals we require the
absence of logarithmic divergencies for the solutions in certain kinematical limits. This
requires the linear combinations of master integrals multiplying the corresponding d ln
terms in the differential equations to vanish in the respective limit. This completely fixes
the remaining boundary terms, i.e. the unknown functions of x respectively the constants.
Since we consider also non-planar integrals it is unavoidable to deal with cuts in s, t
and u at the same time, i.e. we have to handle uncrossed and crossed kinematics. From
the Feynman parameter representation it is clear that there is a Euclidean region with s,
t, u, m2 all less than zero, such that the integrals are real. From the on-shell relation (2.4)
one sees, however, that it is not possible to parametrize this region employing real valued
parameters x, z and m2. Note that in the scheme [34] employed here, the solutions develop
explicit and implicit imaginary parts already during the iterative integration procedure.
We require regularity of each integral in some of the following collinear and, depending
on its cut structure, threshold limits:
z → x, z → 1/x, z → −1, z → (1 + x+ x2)/x, x→ 1 . (4.5)
We emphasize that we impose these conditions for points in the unphysical region, the
algebraically equivalent limits in the physical region may actually be divergent due to
branch cuts. The difference between the two cases lies in the way the signs of the imaginary
parts of the parameters needs to be chosen when approaching the respective point, as
dictated by the Feynman propagator i0 prescription.
We assign a small positive imaginary part to s, t, u and m2 to fix branch cut ambi-
guities. While the m2 dependence is not explicite in our dimensionless master integrals
~m(ε;x, z) anymore, we anticipate its former presence with <(m2) < 0. This translates to
(small) imaginary parts =(x) > 0, =(z) < 0 and =(y) = =((1+x2−xz)/x) > 0. The limits
were computed using in-house Mathematica packages for multiple polylogarithms [61, 62],
where we employed both, coproduct based and non-coproduct based limit algorithms. In
practice we employ small but finite imaginary parts, such that the complex parameters ful-
fil the on-shell relation (2.4). We match the constants appearing in the limit computations
by a numerical fitting procedure. This step utilizes the numerical evaluation of multiple
polylogarithms [63] in GiNaC [43].
5 Solutions and checks
We obtain the solutions in terms of GHPLs of argument z and weights {0,−1, x, 1/x, (1 +
x2)/x, x/(1 +x+x2), (1 +x+x2)/x} and GHPLs of argument x and weights {0,−1, 1, [1 +
– 15 –
JHEP06(2014)032
o2], [1 + o+ o2]}. The explicit expressions are rather lengthy and therefore provided via an
ancillary file on the arXiv only.
We performed several checks on the results. First of all, we integrated the whole
75× 75 system of differential equations at once, fixing consistently all boundary conditions
using the limits described above. We explicitly verified that our solutions fulfil the partial
differential equations both in x and in z. This is a non-trivial check for integrals depending
on both variables, for which we fixed x-dependent boundary terms by regularity conditions.
As a subset of the integrals considered here, we re-calculated all non-trivial planar
master integrals presented in [18]. Taking z′ → z′ + i0 and swapping masters 3 and 4
of sector B213 in the result of that reference, we translate these expressions to our new
functional basis and find perfect agreement at the analytical level.
For the previously unknown non-planar master integrals we compared our re-
sults against numerical samples obtained with the sector decomposition program
SecDec2 [64, 65]. We found the program particularly useful since it allowed us to perform
checks of our results both in the Euclidean and in the physical region. In the Euclidean
region we set x to a truly complex number. Note that to obtain a real number at all consists
already in a very non-trivial check of our solution. In particular, we could verify all our
master integrals in the Euclidean region with a typical precision of at least 4 digits for the
weight 4 coefficients. On the other hand, the numerical evaluation by sector decomposition
in the Minkowski region was much more cumbersome. Using SecDec2 we could evaluate all
corner integrals (integrals with no dots nor scalar products) up to weight 4, finding good
agreement with our result. For sectors involving more than one master integral we addi-
tionally considered integrals with dots and/or scalar products. For them we could check at
least the weight 3 contributions, in some cases also the weight 4 parts. The combination
of these checks in the Euclidean and in the Minkowski region provides stringent evidence
for the correctness of our results.
6 Real valued functions and expansions
For the purpose of numerical evaluation in the physical region the primary form of our solu-
tions is not optimal yet, e.g. because the multiple polylogarithms are not single valued and
their numerical evaluation is not straightforward. We follow the procedure described in [37]
and project onto a new functional basis which consists of Li2,2, classical polylogarithms Lin(n = 2, 3, 4) and logarithms. The Li-functions are related to the G-functions via
Lin(x1) = −G(0, · · · , 0, 1︸ ︷︷ ︸n
;x1) , Li2,2(x1, x2) = G
(0,
1
x1, 0,
1
x1x2; 1
). (6.1)
In our new functional basis we allow for rather complicated rational functions of x and z.
We choose them such that the functions are real valued and the imaginary parts of the
solutions are explicit over the entire physical domain.
In [40] it was demonstrated that this method works also in the presence of generalized
weights, which could in fact be eliminated at the level of the amplitude. In the present case
we work at the level of the master integrals. Also here, we successfully apply this projection
– 16 –
JHEP06(2014)032
onto real valued functions and eliminate all generalized weights {[1 +o2], [1 +o+o2]} using
a coproduct based algorithm.
We can actually go one step further and restrict the target function space even more.
For the functions Lin(x1), Li2,2(x1, x2) we select real arguments with
|x1| < 1 , |x1x2| < 1 . (6.2)
In this way, the multiple polylogarithms are not only real valued but correspond directly
to a convergent power series expansion
Lin(x1) = −∞∑j1=1
xj11jn1, (6.3)
Li2,2(x1, x2) =∞∑j1=1
∞∑j2=1
xj11(j1 + j2)2
(x1x2)j2
j22(6.4)
see e.g. eq. (20) of [63]. While it is not a priori obvious that such a restricted set of
functions is sufficient to represent our master integrals, we find that this is indeed the case.
Our choice of functions drastically improves the numerical evaluation time, since it avoids
additional transformations which would be required otherwise to map to an appropriate
expansion. Evaluating all master integrals discussed in this paper takes only fractions of a
second in a generic phase space point on a single core.
For completeness, we also expand our solutions both at the production threshold and
in the small mass region. The threshold region is characterized by β → 0 for fixed cos θ,
where β =√
1− 4m2/s is the velocity of each vector boson and θ the scattering angle in
the center-of-mass frame, such that z = 1 + 2β(β + cos θ)/(1− β2). We find it convenient
to directly expand our full solutions in the real-valued function representation, rather than
the individual G-functions. The expansion contains β, lnβ, GHPLs of argument cos θ and
weights {−1, 1} as well as the constants ln(2) and Li4(1/2). Similarly, we consider the
small mass limit m2/s → 0 for fixed φ = −(t − m2)/s. The expansion contains m2/s,
ln(m2/s) and GHPLs of argument φ and weights {0, 1}. The first couple of orders for both
expansions as well as our results in terms of real-valued functions are provided via ancillary
files on arXiv.
7 Conclusions
In this paper, we computed the full set of master integrals relevant to the two-loop QCD
corrections to the production of two vector bosons of equal mass in the collision of massless
partons. These two-loop four-point functions are computed using the differential equation
method [24, 28–30]. We describe in detail how we find a canonical basis [31] for the master
integrals. In this basis, the differential equations for the master integrals can be solved in
an elegant and compact manner in terms of iterated integrals. These general solutions are
then matched onto appropriate boundary values, requiring non-trivial transformations of
the iterated integrals. Our analytical results for all master integrals are expressed in terms
of multiple polylogarithms, they are provided with the arXiv submission of this article. We
– 17 –
JHEP06(2014)032
find that it is possible to employ a restricted set of multiple polylogarithms, which allows
for a particularly fast and precise numerical evaluation. We validated our solutions against
numerical samples obtained using sector decomposition.
With the full set of master integrals derived in this paper, it is now possible to derive
the two-loop corrections to the amplitudes for qq →W+W− and qq → ZZ, and to compute
the NNLO corrections to the pair production of massive vector bosons. Combined with
precision measurements of these observables at the LHC, these results will allow for a
multitude of tests of the electroweak theory at unprecedented precision.
Acknowledgments
We are grateful to Sophia Borowka and Gudrun Heinrich for their assistance with SecDec2
and to Pierpaolo Mastrolia for interesting comments on the manuscript. AvM would like to
thank Stefan Weinzierl for solving issues with the GiNaC implementation of the multiple
polylogarithms and Andrea Ferroglia for useful discussions. We acknowledge interesting
discussions with Johannes Henn and Pierpaolo Mastrolia on the properties of the canoni-
cal basis. Finally we thank Kirill Melnikov for comparison of numerical results from [66]
prior to publication. This research was supported in part by the Swiss National Science
Foundation (SNF) under contract PDFMP2-135101 and 200020-149517, as well as by the
European Commission through the “LHCPhenoNet” Initial Training Network PITN-GA-
2010-264564 and the ERC Advanced Grant “MC@NNLO” (340983). The work of AvM was
supported in part by the Research Center Elementary Forces and Mathematical Founda-
tions (EMG) of the Johannes Gutenberg University of Mainz and by the German Research
Foundation (DFG).
A Canonical basis
As a result of the algorithm described in section 3 we find the following canonical basis,
which for simplicity is also attached to the arXiv submission of this paper:
m1 = ε2m2(1+x)2
xfA381 , m2 = ε2m2fA134
2 , m3 = ε2m2z fA1483 , m4 = ε2
m2(1+x2−xz)x
fA1484 ,
m5 = ε2m4(1+x)4
x2fA995 , m6 = ε2
m4(1+x)2
xfA1956 , m7 = ε2m4fA387
7 , m8 = −ε2m4z fA3948 ,
m9 = −ε2 m4(1+x2−xz)
xfA3949 , m10 = ε2m4z2fA408
10 , m11 = ε2m4(1+x2−xz)2
x2fA40811 ,
m12 = ε2m4fA41812 , m13 = −ε3 m
2(1+x)2
xfA5313 , m14 = ε3m2(1+z)fA142
14 ,
m15 = ε3m2(1+x+ x2−xz)
xfA14215 , m16 = −ε3m2(1+z)fA149
16 , m17 = −ε3 m2(1+x+ x2−xz)
xfA14917 ,
m18 = ε3m2(1−x2)
xfA16619 , m19 = ε2
[(1−2ε)(1−3ε) fA166
18 +m2(1+x)2
2xfA381
]− ε3
m2(1+x)
xfA16619 ,
m20 = ε3m2(1−x2)
xfA19821 , m21 = ε2
[(1−2ε)(1−3ε) fA198
20 −m2 fA1342
]− ε3
m2(1−2x2)
xfA19821 ,
m22 = ε3m4(1−x)(1+x)3
x2fA22722 , m23 = ε3
m4(1−x2)
xfA41923 , m24 = ε4
m2(1−x2)
xfA19924 ,
– 18 –
JHEP06(2014)032
m25 = ε4m2(1+z)fA39825 , m26 = ε4
m2(1+x+ x2−xz)x
fA39826 , m27 = ε4
m2(1−x2)
xfA42227 ,
m28 = ε3m4z (1+x)2
xfA17428 , m29 = ε3(1−2ε)
m2(1−x2)
xfA18129 m30 = ε3
m4z (1+x)2
xfA18130 ,
m31 = ε3(1−2ε)m2(1−x2)
xfA18131 m32 = ε3
m4(1+x)2(1+x2−xz)x2
fA18132 ,
m33 = ε4m2(1+x+ x2−xz)
xfA18233 , m34 = ε3m4(1+z)fA182
34 , m35 = ε4m2(1+z)fA18235 ,
m36 = ε3m4(1+x+ x2−xz)
xfA18236 , m37 = ε3
m4z (1+x)2
xfA21437 , m38 = ε3
m4(1+x)2(1+x2−xz)x2
fA21438 ,
m39 = ε3m6z (1+x)2
xfA42739 , m40 = ε4
m4(1+z)(1+x)2
xfA21540 , m41 = ε4
m4(1+x+ x2−xz)(1+x)2
x2fA21541 ,
m42 = ε4m4(z(1+x+ x2) − x)
xfA43042 , m43 = ε4
m6z (1+x)4
x2fA24743 ,
m44 = −ε2m2(1+x)2
2xzfA381 + ε2
5m2
2 zfA1342 + ε2
9m2
2fA1483 + ε3
6m2(1+z)
zfA14916 + ε2
4(1 − 2ε)(1 − 3ε)
zfA16618
− ε32m2(1+x)2
xzfA16619 + ε3
2m4(1+x)2
xfA18130 + ε4
6m2(1+x+ x2−xz)xz
fA18233 + ε3
4m4(1+z)
zfA18234
+ ε4m4(1+x)4
x2fA24744 ,
m45 = ε4m4(1−x)(1+x)3
x2fA24745 , m46 = ε4
m6z2(1+x)2
xfA44646 ,
m47 = ε4m4z
((1+x)2
xfA43042 + (1+z)fA446
47
),m48 = ε3
m4(x− z)(1−xz)x
fB17448 ,
m49 = ε3m4(x− z)(1−xz)
xfB17449 , m50 = ε4
m2(1+x)2
xfB18250 , m51 = ε4
m2(1−x2)
xfB21351 ,
m52 = ε3m4(1+x+ x2−xz)
xfB21352 , m53 = ε3m4(1+z)fB213
53 , m54 = ε3m4(x− z)(1−xz)
xfB21354 ,
m55 = ε3m6(x− z)(1−xz)
xfB24955 , m56 = ε4
m4(1+z)(1+x2−xz)x
fB21556 ,
m57 = ε4m4z (1+x+ x2−xz)
xfB21557 , m58 = ε4
m6(x− z)(1−xz)(1+x2−xz)x2
fB24758 ,
m59 = −ε2 3m2(1+x+ x2−xz)2(x− z)(1−xz) fA134
2 − ε23m2z (1+x+ x2−xz)
4(x− z)(1−xz) fA1483
− ε23m2(1+x2−xz)(1+x+ x2−xz)
4x(x− z)(1−xz) fA1484 − ε3
m4(1+x+ x2−xz)x
fB17449
+ ε43m2(1+x)2(1+x+ x2−xz)
x(x− z)(1−xz) fB18250 + ε4
m4(1+x2−xz)(1+x+ x2−xz)x2
fB24759 ,
m60 = ε4m4(1−x2)2
x2fC23160 , m61 = ε4m4(1+z)2fC252
61 , m62 = ε4m4(1+x)4
x2fC31862 ,
m63 = ε4m4(1+x)2
xfC12663 , m64 = ε4m2(1+z)
[fC12664 − fA182
33
], m65 = ε4
m4(1+x)2
xfC20765 ,
m66 = +ε2m2(1+x)2
4x(1+z)
[fA1342 + z fA148
3
]+ ε3
m2(1+x)2
xfA14916 − ε4
m4(1+x)2(1+x2−xz)x2
fC20765
+ ε4m2(1+x+ x2−xz)
xfC20766 ,
m67 = +ε2m2(1+x)2
4(1+x+ x2−xz)
[fA1342 +
(1+x2−xz)x
fA1484
]+ ε3
m2(1+x)2
x
[fA14917 + ε fA182
33
]− ε4m2(1+z)
[fB21351 + fC207
66 − fC20767
],
m68 = +ε4m2(1−x2)
x
[fC20768 − fA182
33
],
m69 = +ε4m4(1−x2)(1+z)
xfC20765 − ε4
m4(1−x2)(1+x)
x2fC23160 − ε4
m4(1−x2)(1+x)2
x2
[fA21541 − fC239
70
],
– 19 –
JHEP06(2014)032
m70 = + ε4m4(1+x)(1+z)fC20765 + ε4
m4(1+x)2(1−xz)2x2
[fC23160 − 2 fC239
70
]+ ε4
m6(1+x)2(1−xz)(x− z)
2x2fC23969 − ε4
m4(1+x)3
xfA21541 ,
m71 = + ε4m4(1+z)[fA43042 + zfC254
72
]+ ε2
3m2z (1+x)2
2(1−xz)(x− z)fA1342 + ε2
3m2z2(1+x)2
4(1−xz)(x− z)fA1483
+ ε3m4z (1+x)2
x
[fB17448 + εfC207
65
]+ ε2
3m2z (1+x)2(1+x2−xz)4(1−xz)(x− z)x
fA1484 − ε4
3m2z (1+x)4
x (1−xz)(x− z)fB18250 ,
m72 = − ε4m4(1−xz)(x− z)
x (1+z)
[fA43042 + z fC254
72
]− ε3
2m2z (1+x)2
(1+z)x
[fA14214 + fA149
16 − fA14917
]− ε4
2m2z(1+x)2
(1+z)x
[fA39825 − fA182
35 + fB21351 − fC126
64 + fC20766 − fC207
67
]− ε2
3m2z2(1+x)2
2(1+z)2xfA1483
− ε3m4z (1+x)2
(1+z)x
[fA18234 − fB213
53
]− ε4
m4z (1+x)2
2xfC25261 − ε2
m2z (1+x)2(3 + 2x− 4xz + 3x2)
2(1+z)2(1+x+ x2−xz)x fA1342
+ ε4m6z (1+x)2(1−xz)(x− z)
2(1+z)x2fC25471 + ε2
m2z (1+x)2(1+x2−xz)2x (1+z)(1+x+ x2−xz)f
A1484
+ ε4m4(1+x)2(−z + 3x+ 2xz + xz2−x2z)
2(1+z)x2
[fC12663 − fC207
65
],
m73 = − ε4m4(1+x)2z
x
[fC12663 − fC382
74
]+ε2
m2(1+x)2(1+x2)
4(1+x2−xz)x
[fA381 + 4 ε fA166
19
]−ε3m
4(1+x)2(1+x2)
x2fA18132
− (1−2ε)(1−3ε)ε22(1+x2)
(1+x2−xz)fA16618 − ε2
5m2(1+x2)
4(1+x2−xz)fA1342 − ε2
9m2(1+x2)
4xfA1484
− ε3m2(1+x2)(1+x+ x2−xz)
(1+x2−xz)x
[3 fA149
17 + 2m2fA18236
]− ε4
3m2(1+x2)(1+z)
(1+x2−xz) fA18235 ,
m74 = − ε4m4(1+x)2(1−2xz + x2)
x2fC12663 + ε4
m6(1+x)4(1+x2−xz)x3
fC38273
− ε3m4(1+x)2(1+x2)
x2
[2fA181
30 −fA18132 +ε fC382
74
]+ε2
m2(1+x)2(1+x2)(2−3xz+2x2)
4(1+x2−xz)x2z
[fA381 +4εfA166
19
]− ε2
9(1+x2)m2
4x
[2 fA148
3 −fA1484
]− (1−2ε)(1−3ε)ε2
2(1+x2)(2 − 3xz + 2x2)
(1+x2−xz)xz fA16618
− ε25(1+x2)(2 − 3xz + 2x2)m2
4(1+x2−xz)xz fA1342 − ε4
6(1+x2)(1+x+ x2−xz)m2
x2zfA18233
+ ε3(1+x2)(1+x+ x2−xz)m2
(1+x2−xz)x
[3 fA149
17 + 2m2 fA18236
]− ε3
(1+x2)(1+z)m2
xz
[6fA149
16 + 4m2 fA18234
]+ ε4
3(1+x2)(1+z)m2
(1+x2−xz) fA18235 ,
m75 = − ε23m2z
4(1+x)fA1483 + ε2
3m2(2 + x+ xz + 2x2−x2z + x3)
4(1−2ε)(1+x)xfA1484
+ ε2m2(1+x+ 4xz + x2−x2z + x3)
4(1−2ε)(1+x)(1+x2−xz) fA1342 − ε3
(7 + 7x+ 3xz + 3x2−3x2z + 3x3)
(1+x)(1+x2−xz) fA16618
+ ε2(1+x+ xz + x2−x2z + x3)
(1+x)(1+x2−xz) fA16618 − ε4
3m2(1+x+ x2−xz)(1+x)x
fA18233
+ ε3m4(1+x+ x2−xz)(3 + x+ xz + 3x2−x2z + x3)
2(1−2ε)(1+x)(1+x2−xz)x fA18236
+ ε33m2(1+x+ x2−xz)(1+x+ xz + x2−x2z + x3)
2(1−2ε)(1+x)(1+x2−xz)x fA14917 − ε4
m2(1−x2)
xfC38275
− ε3m4(1+z)
(1+x)fA18234 + ε4
12
(1+x2−xz)fA16618 + ε4
2m2(1+x)
(1−2ε)xfA5313 − ε3
9m2
2(1−2ε)xfA1484
– 20 –
JHEP06(2014)032
− ε4m4(1+x)z
xfC38274 + ε3
m2(1+x)(2 + x− xz + x2)
2(1−2ε)(1+x2−xz)x fA381
+ ε3m4(1+x)(1+x+ xz + x2−x2z + x3)
2(1−2ε)x2fA18132 − ε4
6m2(1+x+ x2−xz)(1−2ε)(1+x2−xz)xf
A14917
− ε44m4(1+x+ x2−xz)(1−2ε)(1+x2−xz)xf
A18236 − ε2
m2(1+x)(1+x+ x2−x2z + x3)
4(1−2ε)(1+x2−xz)x fA381
+ ε4m4(1+x)(1+z)
xfC12663 − ε3
m2(1+x)2
2(1−2ε)xfA5313 + ε4
2m2(1+x)2
(1−2ε)(1+x2−xz)xfA16619
− ε42m4(1+x)2
(1−2ε)x2fA18132 + ε4
m4(1+x)3
2x2fC31862 − ε3
m2(1+x)(1+x2)
(1−2ε)(1+x2−xz)xfA16619
− ε52m2(1+3x+ 2xz + x3z − x4)
(1−2ε)(1+x2−xz)x fA18235 + ε4
3m2(1+x2)(1+z)
(1−2ε)(1+x)(1+x2−xz)fA18235
− ε3m2(2 + 5x+ 3xz − 3x2)
2(1−2ε)(1+x)(1+x2−xz)fA1342 .
We remark here that even if the formulas look in some cases rather cumbersome, they
are always at most linear combinations of the starting basis fj with rational coefficients.
Obviously, choosing differently this starting basis can simplify or even complicate substan-
tially these relations. On the other hand the main point of the derivation given in section 3
is to show how, starting from a basis whose differential equations fulfil some initial require-
ments, a canonical basis (if existing!) can be built in an almost algorithmic way.
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
References
[1] J. Ohnemus, Order αs calculations of hadronic W±γ and Zγ production, Phys. Rev. D 47
(1993) 940 [INSPIRE].
[2] U. Baur, T. Han and J. Ohnemus, QCD corrections to hadronic Wγ production with